Dynamics of rigid bodies in a two dimensional incompressible perfect fluid

Olivier Glass (1) , Christophe Lacave (2) , Alexandre Munnier (3, 4) , Franck Sueur (5)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 IF - Institut Fourier
3 IECL - Institut Élie Cartan de Lorraine
4 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
5 IMB - Institut de Mathématiques de Bordeaux

We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore on the fluid pressure. This reformulation highlights geodesic aspects linked to the added mass effect, gyroscopic features generalizing the Kutta-Joukowski-type lift force, including body-body interactions through the potential flows induced by the bodies' motions, body-body interactions through the irrotational flows induced by the bodies' circulations, and interactions between the bodies and the fluid vorticity.

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Glass, Christophe Lacave, Alexandre Munnier, Franck Sueur. Dynamics of rigid bodies in a two dimensional incompressible perfect fluid. Journal of Differential Equations, 2019, 267 (6), pp.3561-3577. ⟨10.1016/j.jde.2019.04.017⟩. ⟨hal-02024104v2⟩